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Coupled Schroedinger equations for diatomic molecules internal motion solved using best adiabatic approximation to obtain energy correction to Born-Oppenheimer method
An adiabatic scheme for separation of the three-dimensional (3D) nuclear dynamics on the ground electronic Born-Oppenheimer potential energy surface of an H 2 O molecule in hyperspherical coordinates is presented. It is found that the three vibrational modes are weakly coupled and the 3D vibrational wave function can be approximated as a product of three separable functions: one represented by the hyperradius and two by the two hyperangles individually. Additionally, this framework is then used for investigation of the formation and the role of a saddlelike barrier arising in the two hyperspherical angles that is to moderate the OH + H dissociation process. In order to test the validity of the framework, vibrational states with energies up to 19 500 cm -1 are constructed under the assumptions of adiabaticity and separability and compared to full three-dimensional high-precision numerical calculations yielding remarkable correspondence. As a result we present a simple construction scheme for separated molecular vibration states as the first step towards theoretical investigation of laser-driven molecular dynamics of triatomic molecules.
Adiabatic approximation for nonreactive molecular collisions obtained from variational principle
Topological flat moiré bands with nearly ideal quantum geometry have been identified in homobilayer transition metal dichalcogenide moiré superlattices, and are thought to be crucial for understanding the fractional Chern insulating states recently observed therein. Previous study proposed viewing the system using an adiabatic approximation that replaces the position dependence of the layer spinor with a nonuniform periodic effective magnetic field. When the local zero-point kinetic energy of this magnetic field cancels identically against that of an effective Zeeman energy, a Bloch-band version of Aharonov-Casher zero-energy modes, which we refer to as Aharonov-Casher band, emerges leading to ideal quantum geometry. Here, we critically examine the validity of the adiabatic approximation and identify the parameter regimes under which Aharonov-Casher bands emerge. We show that the adiabatic approximation is accurate for a wide range of parameters including those realized in experiments. Furthermore, we show that while the cancellation leading to the emergence of Aharonov-Casher bands is generally not possible beyond the leading Fourier harmonic, the leading harmonic is the dominant term in the Fourier expansions of the zero-point kinetic energy and Zeeman energy. Finally, the leading harmonic expansion accurately captures the trend of the bandwidth and quantum geometry, although it may fail to quantitatively reproduce more detailed information about the bands such as the Berry curvature distribution.
Ground state energy of molecule in adiabatic approximation
A simple formula for the transition probability for electron exchange between unlike ions and atoms is established within the adiabatic approximation by employing the Linear Combination of Atomic Orbitals (LCAO) method. The formula also involves an adiabatic parameter, introduced by Massey, and thus the difficulties arising from the internal energy defect and the adiabatic approximation are avoided. Specific reactions Li(+++) + H to Li(++) + H(+) and Be(4+) + H to Be(3+) + H(+) are considered as examples. The calculated capture cross section results of the present work are compared with the experimental data and with the calculation of other authors over the velocity range of 10(7) cm/sec to 10(8) cm/sec.
Eliashberg–McMillan theory of superconductivity is essentially based on the adiabatic approximation. Small parameter of perturbation theory is given by λ((Ω{sub 0})/(E{sub F})) ≪ 1, where λ is the dimensionless electron–phonon coupling constant, Ω{sub 0} is characteristic phonon frequency, while E{sub F} is Fermi energy of electrons. Here we present an attempt to describe the electron–phonon interaction within Eliashberg–McMillan approach in situation, when characteristic phonon frequency Ω{sub 0} becomes large enough (comparable to, or exceeding, the Fermi energy E{sub F}). We consider the general definition of electron–phonon pairing coupling constant λ, taking into account the finite value of phonon frequency. Also, we obtain the simple expression for generalized coupling constant λ-tilde that determines the mass renormalization, with the account of finite width of conduction band, which describes the smooth transition from the adiabatic regime to the region of strong nonadiabaticity. In the case of strong nonadiabaticity, when Ω{sub 0} ≫ E{sub F}, the new small parameter appears, λ((E{sub F})/(Ω{sub 0})) ~ λ(D/(Ω{sub 0})) ≪ 1 (D is conduction band half-width), and corrections to electronic spectrum become irrelevant. At the same time, the temperature of superconducting transition T{sub c} in antiadiabatic limit is still determined by Eliashberg–McMillan coupling constant λ, while the preexponential factor in the expression for T{sub c}, conserving the form typical of weak-coupling theory, is determined by the bandwidth (Fermi energy). For the case of interaction with a single optical phonon, we derive the single expression for T{sub c}, valid both in adiabatic and antiadiabatic regimes and describing the continuous transition between these two limiting cases. The results obtained are discussed in the context of superconductivity in FeSe/STO.
A counterexample to the adiabatic approximation theorem is given when degeneracies are present. A formulation of an alternative version is proposed. A complete asymptotic decomposition for n dimensional self-adjoint Hamiltonian systems is restated and used.
A decomposition of the exact exchange–correlation potential of time-dependent density functional theory into an interaction component and a kinetic component offers a new starting point for non-adiabatic approximations. The components are expressed in terms of the exchange–correlation hole and the difference between the one-body density matrix of the interacting and Kohn–Sham systems, which must be approximated in terms of quantities accessible from the Kohn–Sham evolution. We explore several preliminary approximations, evaluate their fulfillment of known exact conditions, and test their performance on simple model systems for which available exact solutions indicate the significance of going beyond the adiabatic approximation.
Nonrelativistic energy of excited vibrational state of molecule calculated in adiabatic approximation shown to be upper bound to exact nonrelativistic energy
Time-dependent density functional theory continues to draw a large number of users in a wide range of fields exploring myriad applications involving electronic spectra and dynamics. Although in principle exact, the predictivity of the calculations is limited by the available approximations for the exchange-correlation functional. In particular, it is known that the exact exchange-correlation functional has memory-dependence, but in practise adiabatic approximations are used which ignore this. Here we review the development of non-adiabatic functional approximations, their impact on calculations, and challenges in developing practical and accurate memory-dependent functionals for general purposes.
A theory for the pitch-angle scattering of cosmic rays in a turbulent magnetic field can be derived from first principles by applying both the quasi-linear and adiabatic approximations to the master equation for the ensemble averaged distribution function. A proof of the failure of these approximations, taken together, is given. Some predictions of the quasi-linear theory, are given. New wave-like propagation modes have been discovered.
An adiabatic approximation is applied to the calculation of turbulent MHD dynamo magnetic fields in thin disks. The adiabatic method is employed to investigate conditions under which magnetic fields generated by disk dynamos permeate the entire disk or are localized to restricted regions of a disk. Two specific cases of Keplerian disks are considered. In the first, magnetic field diffusion is assumed to be dominated by turbulent mixing leading to a dynamo number independent of distance from the center of the disk. In the second, the dynamo number is allowed to vary with distance from the disk's center. Localization of dynamo magnetic field structures is found to be a general feature of disk dynamos, except in the special case of stationary modes in dynamos with constant dynamo number. The implications for the dynamical behavior of dynamo magnetized accretion disks are discussed and the results of these exploratory calculations are examined in the context of the protosolar nebula and accretion disks around compact objects.
When a system has evolved far from a ground-state, the adiabatic approximations commonly used in time-dependent density functional theory calculations completely fail in some applications, while giving qualitatively good predictions in others, and sometimes even quantitative predictions. It is not clearly understood why this is so, and developing practical approximations going beyond the adiabatic approximation remains a challenge. This paper explores three different lines of investigation. First, an expression for the exact time-dependent exchange–correlation potential suggests that the accuracy of an adiabatic approximation is intimately related to the deviation between the natural orbital occupation numbers of the physical system and those of the Kohn–Sham system, and we explore this on some exactly-solvable model systems. The exact expression further suggests a path to go beyond the adiabatic approximations, and in the second part we discuss a newly proposed class of memory-dependent approximations developed in this way. Finally, we derive a new expression for the exact exchange–correlation potential from a coupling-constant path integration.
By integrating the exact equations of motion, particle orbits have been followed in a good model magnetospheric field consisting of a planetary dipole, forward magnetosphere, and magnetotail current system. Proton energies from 2 eV to 20 keV were used for the full range of equatorial pitch angles and phase. Despite considerable pitch angle scattering in the equatorial plane crossings, it is found, first, that the bounce-averaged cross-tail drift velocity is approximately independent of pitch angle. Second, it is found that, averaged over initial gyrophase, the drift velocity (due to field curvature and gradient) is proportional to proton energy and is given to good approximation by adiabatic approximations, even up to 20 keV, despite the extreme lack of meeting the adiabatic criteria.
A new transport equation for the cosmic-ray omnidirectional intensity is obtained. This equation follows exactly from the coupled pair of differential moment equations we presented earlier. It can be characterized as a nonlocal convection-diffusion equation in which the usual transport coefficients are replaced by time integral operators. The nonlocal equation is shown to reduce to the standard convection-diffusion form if the adiabatic approximation can be applied. In general, the adiabatic approximation does not apply; however, by going to the limits of infinite and zero gyroradius and, in addition, applying the adiabatic approximation, the large- and small-gyroradius transport theories due originally to Jokipii are regained. The validity of these theories as asymptotic limits and as approximate theories in the interplanetary magnetic field is discussed.
Adiabatic exchange equation in electron scattering from multi-electron atoms - elastic scattering cross sections for sodium and lithium
The equation which governs the quasi-linear approximation to the ensemble and gyro-phase averaged one-body probability distribution function is constructed from first principles. This derived equation is subjected to a thorough investigation in order to calculate the possible limitations of the quasi-linear approximation. It is shown that the reduction of this equation to a standard diffusion equation in the Markovian limit can be accomplished through the application of the adiabatic approximation. A numerical solution of the standard diffusion equation in the Markovian limit is obtained for the narrow parallel beam injection. Comparison of the diabatic and adiabatic results explicitly demonstrates the failure of the Markovian description of the probability distribution function. Through the use of a linear time-scale extension the failure of the adiabatic approximation, which leads to the Markovian limit, is shown to be due to mixing of the relaxation and interaction time scales in the presence of the strong mean field.